ODE
\[ 3 y'(x)=\sqrt {x^2-3 y(x)}+x \] ODE Classification
[[_1st_order, _with_linear_symmetries], _dAlembert]
Book solution method
Exact equation, integrating factor
Mathematica ✓
cpu = 0.144187 (sec), leaf count = 547
\[\left \{\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,1\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,2\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,3\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,4\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,5\right ]\right \},\left \{y(x)\to \text {Root}\left [3888 \text {$\#$1}^6-1944 \text {$\#$1}^5 x^2+243 \text {$\#$1}^4 x^4+216 \text {$\#$1}^3 e^{6 c_1}-378 \text {$\#$1}^2 e^{6 c_1} x^2+144 \text {$\#$1} e^{6 c_1} x^4-16 e^{6 c_1} x^6+3 e^{12 c_1}\& ,6\right ]\right \}\right \}\]
Maple ✓
cpu = 0.091 (sec), leaf count = 131
\[ \left \{ -{\frac {1}{y \left ( x \right ) } \left ( \left ( x-\sqrt {{x}^{2}-3\,y \left ( x \right ) } \right ) \sqrt {2\,\sqrt {{x}^{2}-3\,y \left ( x \right ) }+x}+\sqrt {2\,\sqrt {{x}^{2}-3\,y \left ( x \right ) }-x}\sqrt {-{x}^{2}+4\,y \left ( x \right ) }y \left ( x \right ) {\it \_C1}\, \left ( x+\sqrt {{x}^{2}-3\,y \left ( x \right ) } \right ) \right ) {\frac {1}{\sqrt {2\,\sqrt {{x}^{2}-3\,y \left ( x \right ) }-x}}}{\frac {1}{\sqrt {-{x}^{2}+4\,y \left ( x \right ) }}} \left ( x+\sqrt {{x}^{2}-3\,y \left ( x \right ) } \right ) ^{-1}}=0 \right \} \] Mathematica raw input
DSolve[3*y'[x] == x + Sqrt[x^2 - 3*y[x]],y[x],x]
Mathematica raw output
{{y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 1]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 2]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 3]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 4]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 5]}, {y[x] -> Root[3*E^(12*C[1]) - 16*E^(6*C[1])*x^6 + 144*E^(6*C[1])*x^4*#1 - 378*E^(6*C[1])*x^2*#1^2 + 216*E^(6*C[1])*#1^3 + 243*x^4*#1^4 - 1944*x^2*#1^5 + 3888*#1^6 & , 6]}}
Maple raw input
dsolve(3*diff(y(x),x) = x+(x^2-3*y(x))^(1/2), y(x),'implicit')
Maple raw output
-1/(2*(x^2-3*y(x))^(1/2)-x)^(1/2)/(-x^2+4*y(x))^(1/2)*((x-(x^2-3*y(x))^(1/2))*(2*(x^2-3*y(x))^(1/2)+x)^(1/2)+(2*(x^2-3*y(x))^(1/2)-x)^(1/2)*(-x^2+4*y(x))^(1/2)*y(x)*_C1*(x+(x^2-3*y(x))^(1/2)))/(x+(x^2-3*y(x))^(1/2))/y(x) = 0